Local nonlinear geometry and estimates

Local nonlinear geometry and estimates

The nonlinear argument replaces a flat Lie-algebra field by actual group-valued links while controlling the difference from the quadratic model. It first constructs the chart and its Haar density, then obtains local inverse and predictor estimates, and finally pays for the exceptional configurations through packet and support budgets. Each estimate records whether it is a pointwise bound, a derivative bound on a healthy domain, or a summable bound over supports. These three forms of control have different uses in the primitive integration and cannot be substituted for one another.

Constructing the exponential chart and its Haar ratio

The chart is constructed from the closed matrix image of the compact gauge group. Its real Lie algebra consists of matrices such that exp() belongs to that image for every real. Let be the orthogonal complement of this Lie algebra in the ambient real matrix space. First suppose 0, exp(, and. For0, put. Then and exp() = (exp exp(exp(

Closedness gives exp(; negative follow by inversion. Thus belongs to the Lie algebra.

Consequently, in some neighborhood of zero in, the condition exp( forces Otherwise choose nonzero such0 and extract a convergent subsequence of their unit directions. Its limit has norm one, belongs to, and by the preceding argument belongs to the Lie algebra, a contradiction.

Identify the color space linearly with the Lie algebra by, and write exp) = exp( The map exp()) exp(

has derivative ( at zero, an isomorphism by the orthogonal decomposition. Its local smooth inverse has a color component log. In particular log(exp( for small a nearby has inverse coordinates (), then exp() = exp(. The isolation just proved gives = 0, so exp((log. Shrinking the color ball, its exponential image is open in: continuity of the inverse sends a group neighborhood of each image point back into the ball. No global logarithm or connectedness assumption is used. For) = log exp())) where the translated point lies in the chart. Put

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A(a) = DFexp(−C(a))(a),
J(a) = det A(a).

Smoothness of the ambient inverse makes smooth, and is the identity germ, so ), the identity of germs near

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Fexp(−C(a)) = Fexp(−C(b)) ◦Fu

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gives, by differentiation and determinants,

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J(b) det DFu(a) = J(a).

Shrink the ball 2 there and put = exp). Push forward by exp obtaining a finite nonzero measure supported on For measurable, the change-of-variables formula and the overlap law give

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v(u−1A) = v(A ∩uU).

Indeed is injective on the relevant chart pieces, and positivity of makes its density identity det

Average over normalized Haar measure. Tonelli and right invariance make the left integral ). For fixed, the set of with has measure), by inversion and right translation, so the right integral is). Both constants are positive and finite: is nonempty and open, and is bounded above and below on the finite-radius ball. Hence 0 satisfies (exp

Intersecting an arbitrary measurable test set with proves the full local measure identity. This gives the required normalization, positivity, smoothness and value

Local Haar charts in the primitive integral

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Active and dormant coordinates; amendment A2.
The chart integral contains rh active
real coordinates and dv dormant Gaussian coordinates. Integration of the latter gives one, including
when dv = 0; it does not authorize dropping their normalization. Put K =
√
2π and mg = c(Kg)r.
The f −h frozen free links cancel as mf
ge−(f−h) log mg = mh
g. Consequently

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K−(rh+dv)mh
g = K−(rh+dv)chKrhgrh = K−dvchgrh.

The extracted exponent would instead produce, an extra factor use the reconciled exponent) below. The determinant of the reference form remains a separate factor. The local change of variables in the primitive expansion uses the chart ratio of the same compact gauge group. If is its chart ball and the local Haar ratio is), then

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a 7−→expG(a)V

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sends 1Bρ(a)cJ(a) da to Haar measure restricted to expG(Bρ)V , for every fixed V ∈G. This
follows by the defining chart measure identity for the exponential followed by right Haar invariance.
On the finite set of active links, take the product of these maps. Product measure and multiplication
of densities give the exact joint density Q
i cJ(ai) on the product chart ball, with the product of
the restricted Haar measures as its image. The inactive links are held fixed.

The active chart is sufficient because the sector selectors force every active link of a nonzero integrand to lie in that chart relative to its replica background. More precisely those links have the form exp with inside the chart ball. Consequently restricting integration to the product chart does not discard a nonzero contribution. The transformed integrand’s powers of and its logarithmic Jacobian terms come from this product density.

Before this change, the tree-gauge configuration measure decomposes into active-link and exterior-link coordinates by a measure-preserving splitting. Fubini permits integration first over the active coordinates and then the exterior coordinates, even after the team’s bounded local ingredients are inserted. Regrouping the integrand after this substitution produces its quadratic Gaussian density and the remaining explicit vertex factors. The chart identity and product-measure operations are exact. The local Haar-chart construction, grafted-signal argument and Gaussian exponent rearrangement in this chapter supply the chart ratio, selector implication and vertex decomposition used here.

The selector-to-chart implication has the following numerical form. On an active link let , with its replica background. The actual sector and tree-slice conditions give chord( where K is the selected shell width and t the small-field scale. The local logarithm estimate giveslog chord() and exp(log on the wider shell chord(. Choose a with and, the chart radius. Then = log satisfies and = exp. This is the chart representation of every active link of a nonzero sector integrand. Its substantive geometric input is the shell-access estimate deriving chord( from the actual sector conditions.

The inserted cutofffunctions can be handled at all color arguments. Write) for the inner- It lies between zero and one, equals one on, and can be share cutoffat radius at. nonzero only on Therefore = 1 wherever is nonzero. Define the extended Jacobian logarithm by) log). On the support of, positivity of J in the chart gives exp(), provided 2. This identity yields the exact product Jacobian in

the integral. Outside the chart the cutoffmakes the transformed integrand vanish, so no exterior logarithm value is used in the change of variables.

Since(0) = 1 and J is smooth and positive near zero, log J is locally Lipschitz with log . Wherever is nonzero,. Hence for all v, after making 4at smaller than the fixed local radius. The zero-cutoffcase is immediate. This proves the uniform small Jacobian-log bound used in the primitive vertex envelopes.

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The later rescaling from active chart variables to fluctuation variables is likewise exact. Let g > 0
be the coupling scale, p the real number of active coordinates, and separate fluctuation coordinates
into active and vacant parts. The active affine dilation used in the source sends gp times Lebesgue
measure to active-chart Lebesgue measure. Its coordinate reassembly is a permutation and preserves
measure. The integrand depends on the active part through the original chart expression; the vacant
part contributes a separate normalized Gaussian comfort factor with integral one. Thus, in these
coordinates, its integral factors as
Z
Z
Z
Y (x)gpC(y)dxdy = gp
Y (x)dx,
C(y)dy = 1.

The dormant quadratic energy identity gives precisely this factorization of the transformed ribbons, cutoffand Jacobian terms. Fubini is licensed by the primitive integrable envelopes. This accounts for the active-coordinate Jacobian and the unused Gaussian coordinates in the exact fluctuation integral, rather than discarding them by a dimension convention. The affine dilation, dormant energy and normalization are derived next; the packet and non-packet estimates below supply the vertex envelopes.

The coordinate map in that rescaling is simply affine dilation followed by reindexing. Index active real coordinates by pairs (active link, color coordinate), let p be their cardinality, and put . Its derivative is g times the identity, so det( 0. The finite-dimensional change-of-variables formula says D pushes dx to Lebesgue measure. Reindex the pairs into one color vector per active link; this is a coordinate permutation followed by regrouping product coordinates and preserves Lebesgue measure on measurable rectangles, hence on all measurable sets. The active coordinate of the original lifted fluctuation is exactly this The configuration, active cutoffproduct and Jacobian-log sum depend only on these active coordinates.

On the vacant coordinates the source’s dormant matrix is the identity and on active coordinates it is zero. Its quadratic energy is therefore one half the sum of the squared vacant real coordinates. If is their number, its comfort density is

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C(y) = (
√
2π)−dv exp(−∥y∥2/2).

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The product of the one-dimensional Gaussian integrals gives
R C = 1, also for dv = 0. This
proves the dormant factor and its normalization used in the previous integral identity.
The remaining volume constants can be checked by actual finite set partitions. If r is the color
dimension, h is the number of active free links, and f is the total number of free tree-gauge links,
then p = rh, the number of dormant coordinates is dv, and the ambient fluctuation dimension is
p + dv. The number of frozen free links is f-h. Define mg = c(
√
2π g)r, where c is the Haar chart
scale. The chart/rescaling prefactor satisfies the exact identity

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(
√
2π)−dvgrhch
= (
√
2π)−(rh+dv)
mf
g exp(−(f −h) log mg).

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It follows by cancelling mf−h
g
and substituting its definition. The determinant of the reference
blank quadratic form contributes the separate factor exp(−1
2 log det Qblank); it is not included in
a dimension count. Together these are the deducted Gaussian normalization used by the local
integral. Its reference covariance has density

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(
√
2π)−d exp(−log det Q/2 −zTQ−1z/2)

in ambient dimension d, with positive definite Q. Thus all determinant and Gaussian-volume factors are explicit before the remainder vertices are estimated. There is also a useful exact cancellation in the raw density before this transformation. Splitting the action into a retained crown and its reduced part gives

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β cutR(U) + ER(U)
= vR|Λ| −reducedAction(U) −additiveActivity(U).

The block-kernel correction has whole gap cut)-coarseCrown( ). Its exponential cancels the retained crown already moved into the raw density, since

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exp(−β coarseCrown) exp(β cutR)
exp(−β(cutR −coarseCrown)) = 1.

Consequently the sector sum of the raw marked density equals the tagged block kernel times the original marked density pointwise. A summable envelope for the original activity gas and bounded sector factors licenses integration and summation. This is the initial action cancellation in the primitive Mark formula. The local vertex decomposition and primitive-integral estimates in the marked-integration chapter complete this formula.

The quadratic floor before choosing the local covariance

The reference precision is a Gram matrix with an identity completion on dormant coordinates. Let L u be the active part of a link direction, let D u be the physical stacked motion, and let be the diagonal projection onto dormant real coordinates. The plate is, where F is the scaled motion matrix. The scale satisfies

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uTF TFu = σ2 X
|Du|2.

To prove a volume-uniform positive floor, the source first obtains at each link the local inequality

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|Lui| ≤ZRi(u) + MCi(u).

Here is the square root of the sum of squared physical motions within a fixed native radius d of i, while is the square root of the sum of squared active link directions within radius 3B. Their definitions use actual torus max-distance and the actual link census. Summing squared inequalities and charging how often any term occurs gives

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W ≤2Z2NdV + 2M2N3BW,

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where W = P |L ui|2, V = P |D u|2 and Nr = (2r + 1)4 Lblock4 |Axis|. The local geometric
driving estimate supplies Z fixed at the chosen block and stage, while M has reference-coupling order
−1/2 up to the displayed width factors. Since N3B has only polynomial width growth, eventually
2M2N3B ≤1/2. Absorption gives W ≤4Z2Nd V. Enlarge the constant to a > 0 if needed.

Every real direction splits orthogonally into its active and dormant components:

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∥u∥2 = ∥Lu∥2 + uTPdu, uTPdu ≥0.
Taking k = min(σ2/a, 1) > 0 proves

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uTQu ≥k∥u∥2.

This floor uses the physical motion estimate, not an assumed lower bound for the full matrix. The local driving and flattening arguments are developed explicitly in (NF1)–(NF4).

The localized inverse-column construction also treats dormant coordinates exactly. The plate has identity row and column on each dormant coordinate, and its linear source there is zero. The same is true in every principal restriction. Its inverse therefore has identity row and column at those coordinates, by multiplying the inverse equation by that row. Padding a chosen principal inverse column into the full space preserves this property; offits footprint it is zero, and the diagonal belongs to the footprint. Symmetrizing these padded columns makes the reference covariance identity on the dormant rows and columns. The mean, defined as minus the transpose of the padded inverse times the linear source, is zero on dormant coordinates. These identities justify the dormant Gaussian factor used earlier. Positivity and the uniform upper bound of the full symmetrized padded covariance additionally require the localized inverse-window estimates.

Exponentially local inverses and a positive finite-window covariance

The covariance has its own construction An exponentially decaying inverse and a positive covariance on a finite window are separate objects. Truncating an inverse kernel need not preserve positivity. The argument below therefore constructs the window covariance with a positive margin and retains the comparison error in the norm used by the Gaussian integral. The later coordinates, mean and determinant refer to this particular covariance. They do not permit substitution of an arbitrary matrix with similar entrywise decay.

Let A be a real symmetric matrix indexed by coordinates placed at sites. Suppose 0, and its weighted absolute row and column sums at width0 are at most 0. All principal restrictions inherit these statements: extend a vector by zero for the quadratic floor, and restrict the nonnegative row sums for the price. Choose 01 with For0 put. It is positive definite and has an actual two-sided inverse. Fix a column j, put = exp( dist()), and set . Conjugation by the diagonal s gives

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(Au + E)w = ej,
Eik = (Au)ik(si/sk −1).

The triangle inequality for distance gives

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|si/sk −1| ≤θ exp(η dist(ai, ak)).

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For completeness, the logarithm of si/sk lies between −θD and θD, where D = η dist(ai, ak) ≥0.
Convexity gives eθD −1 ≤θ(eD −1) ≤θeD. Also 1 −ev ≤e−v −1 when v ≤0. These two bounds
prove the displayed inequality. The diagonal contribution of uI vanishes from E, so its absolute row
and column sums are at most θX ≤r/2.
The elementary Schur quadratic estimate is |vTEv| ≤(r/2)∥v∥2: expand the double sum and
use 2|vivk| ≤v2
i + v2
k, then apply the two sum bounds. Taking the inner product of the conjugated
equation with w yields

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(r/2 + u)∥w∥2 ≤wj.

With = min(12), this implies, hence; the zero-vector case is immediate. Therefore

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exp(θη dist(ai, aj))|(A + uI)−1
ij |

[min(1

This is uniform in matrix dimension and in all principal restrictions.

For the positive-width conclusion now take 0, set = min(1 and 0. The preceding entrywise estimate also holds when = 0; the following exponential site sum uses 0. If at most m coordinate labels occupy each site, the entrywise inverse estimate converts to row and column estimates at width 2k. Indeed the sum of exp() dist) over torus sites is bounded by the corresponding absolutely convergent lattice sum, independently of the period. Multiply that sum by[min(1; call the resulting price P. Thus both the full inverse and every zero-padded principal inverse column have price at most P at width 2k.

For each column j choose a principal window containing all coordinates within distance B of and contained within distance B+2. Its padded inverse column is

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

(A|Wj)−1
ij,
i ∈Wj,
Kij =

0,
i /∈Wj.

The error vanishes exactly when, by the principal inverse equation. In particular it vanishes for dist(. Product price bounds at width 2k give price( there. On its remaining entries, losing one width unit k supplies exp(), so at width k its price is at most

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εB = exp(−kB)(XP + 1).

The resolvent identity then gives width-k price at most. These are bounds on actual matrices, not a declaration that the padded inverse is the full inverse.

Symmetrize2. Since is symmetric, averaging the two error bounds gives price(, and price(. The Schur bound turns the error price into . Also implies. For example, put and use nonnegativity of (), together with. Expanding and using gives

which proves. Consequently, once

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(2X)−1I ≤C, |vTCv| ≤P∥v∥2.

Thus C is the positive covariance required for the reference Gaussian, with the uniform upper bound used in the warning estimate. Its entries vanish beyond distance B+2. The residual ACI has width-k price at most, by multiplying). All these choices are uniform in the torus volume and preserve the original endpoint-defined windows.

For the physical plate, the quadratic floor proved above supplies r, while the stencil bounds supply its width- row price X and the coordinate census supplies m. At fixed block and stage, increasing B makes the exponential error small enough to meet the displayed requirement. The physical operator bounds and local driving estimate of this chapter establish these inputs for the plate.

Axial gauge and bounded logarithmic extension

Axial gauge is used locally to make a controlled logarithm available on a clean hull. Its chosen tree is an auxiliary coordinate device. The proof must therefore bound the resulting logarithm and preserve the observables when it extends links outside the hull. Subsequent comparison estimates depend on the size of the fixed inspected hull, not on a simultaneous global gauge choice for every query.

For a finite subset X of the integer cover, let) be the sum of the chords of its internal positive links and) the sum of the chords of its wholly internal elementary plaquettes. Boundary links and incomplete plaquettes contribute zero. Both sums are nonnegative, and is gauge invariant.

An ordered list l of distinct coordinate axes defines a slab: coordinates in l range over the integersr,. . . ,r, while the other coordinates are pinned at o. From a point p in the slab, travel first in the first listed axis to the corresponding coordinate of x, then in the next axis, and so on. Let) be the ordered product of links along this route. Its endpoint is x because the remaining coordinates are pinned. For l followed by a new axis j, the gauge makes every internal j-link equal to 1. Indeed the routes to x and have the same initial part; their final straight j-segments differ by precisely the link (x,j), so the gauge-transformed link cancels.

When these j-links equal 1, the plaquette relation gives

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| chord U(x + ej, i) −chord U(x, i)|
≤chord holonomy(square(i, j), x).

This follows from the triangle inequality applied to a and its inverse, where a,b are the two horizontal links. For each site the right side is at most the sum of its internal plaquette chords; reversing the ordering of the two axes only inverts the holonomy and leaves its chord unchanged. Summing over the four positive axes and over a slice gives a bound of four times the plaquette sum in the adjacent layer. Telescoping between two slices therefore yields

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|eslice(z)U −eslice(u)U| ≤4fslab(U).

All layers used by the telescoping sum lie inside the slab; adding the unused nonnegative layers proves this bound even when the two slices are far apart. We prove by induction on = length() that some route root p satisfies

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eslab(gauge HpU) ≤4k(2r + 1)fslab(U).

For = 0 the slab is a single site with no internal links. For the induction step choose, among the 2r+1 slices normal to the new axis, one whose internal face sum is minimal. Its face sum is at most + 1), because the slice face sets are disjoint subsets of the full slab face set. Use the induction hypothesis to choose a root on that slice, then extend its routes in the new axis. The edge sum on the chosen slice is unchanged, the new-axis links are 1, and the preceding slice comparison bounds every other slice by the chosen slice edge sum plus 4). Summing all 2r+1 slices gives the required bound with k+1. This also proves existence of the root; no curvature smallness was required.

For all four axes the slab is the box [, so the bound is 16(2r+1)). If every plaquette chord in a periodic radius-r patch is at most q, then |PlaqAxes|q. Lift the patch to that box and apply the preceding construction. When 2, the lift is injective on the box, so descends to a gauge on the corresponding torus sites, extended arbitrarily elsewhere. If